Research on a classification theory of prehomogeneous vector spaces

准齐次向量空间的基本理论首先由佐藤干雄提出，后来许多数学家一直在发展这一理论。这一点已经用数论方法、代数分析方法或表示论方法研究过。有应用理论的zeta函数等，因此这是一个重要的问题，以分类他们知道什么样的例子存在。第一个结果是分类的不可约prehomogeneous向量空间获得的佐藤干雄和木村辰夫。然而，分类问题是非常困难的，一些数学家说是不可能的。但Mikio Sato在非不可约情形下也得到了一些结果。在我们的工作中，我们开发了很多他的成果。我们还从与M不同的观点对一些准齐次向量空间进行了分类。佐藤的研究我们还找到了一些工具，如PV-等价。

The basic theory of prehomogeneous vector spaces was first obtained by Mikio Sato, and later many mathematicians have been developing this theory. This has been investigated by number theoretical method, algebra-analytic methos or representation theoretical method. There are applications to the theory of zeta functions etc. Hence it is an important problem to classify them to know what kind of examples exist.The first result was a classification of irreducible prehomogeneous vector spaces obtained by Mikio Sato and Tatsuo Kimura. However a classification problem is very difficult and some mathematicians say impossible. But Mikio Sato has some result even for non-irreducible case. In our work, we developed a lot his results. We have also classified some prehomogeneous vector spaces with the different view point from M. Sato’s research. We also find some tools such as PV-equivalences.